Optimal. Leaf size=125 \[ \frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^6(c+d x)}{3 d}+\frac{a b \tan ^4(c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.104404, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 948} \[ \frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^6(c+d x)}{3 d}+\frac{a b \tan ^4(c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{b^2 \tan ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 948
Rubi steps
\begin{align*} \int \sec ^8(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(b+a x)^2 \left (1+x^2\right )^2}{x^8} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{b^2}{x^8}+\frac{2 a b}{x^7}+\frac{a^2+2 b^2}{x^6}+\frac{4 a b}{x^5}+\frac{2 a^2+b^2}{x^4}+\frac{2 a b}{x^3}+\frac{a^2}{x^2}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{a^2 \tan (c+d x)}{d}+\frac{a b \tan ^2(c+d x)}{d}+\frac{\left (2 a^2+b^2\right ) \tan ^3(c+d x)}{3 d}+\frac{a b \tan ^4(c+d x)}{d}+\frac{\left (a^2+2 b^2\right ) \tan ^5(c+d x)}{5 d}+\frac{a b \tan ^6(c+d x)}{3 d}+\frac{b^2 \tan ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.654685, size = 104, normalized size = 0.83 \[ \frac{\tan (c+d x) \left (21 \left (a^2+2 b^2\right ) \tan ^4(c+d x)+35 \left (2 a^2+b^2\right ) \tan ^2(c+d x)+105 a^2+35 a b \tan ^5(c+d x)+105 a b \tan ^3(c+d x)+105 a b \tan (c+d x)+15 b^2 \tan ^6(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 110, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) +{\frac{ab}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{b}^{2} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}}+{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12142, size = 123, normalized size = 0.98 \begin{align*} \frac{7 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{2} +{\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} b^{2} - \frac{35 \, a b}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.487539, size = 231, normalized size = 1.85 \begin{align*} \frac{35 \, a b \cos \left (d x + c\right ) +{\left (8 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{6} + 4 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, b^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17727, size = 159, normalized size = 1.27 \begin{align*} \frac{15 \, b^{2} \tan \left (d x + c\right )^{7} + 35 \, a b \tan \left (d x + c\right )^{6} + 21 \, a^{2} \tan \left (d x + c\right )^{5} + 42 \, b^{2} \tan \left (d x + c\right )^{5} + 105 \, a b \tan \left (d x + c\right )^{4} + 70 \, a^{2} \tan \left (d x + c\right )^{3} + 35 \, b^{2} \tan \left (d x + c\right )^{3} + 105 \, a b \tan \left (d x + c\right )^{2} + 105 \, a^{2} \tan \left (d x + c\right )}{105 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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